Some uniform effective results on André–Oort for sums of powers in Cⁿ

We prove an André–Oort-type result for a family of hypersurfaces in Cⁿ that is both uniform and effective. Let K_* denote the single exceptional imaginary quadratic field which occurs in the Siegel–Tatuzawa lower bound for the class number. We prove that, for m, n ∈ Z>0 , there exists an effective constant c(m, n)>0 with the following property: if pairwise distinct singular moduli x₁, …, xₙ with respective discriminants Δ₁, …, Δₙ are such that a₁ x₁ᵐ + ⋯ + aₙ xₙᵐ ∈ Q for some a₁, …, aₙ ∈ Q \0\ and # \ Δᵢ \;:\; Q(√Δᵢ) = K_*\ ≤ 1 , then maxᵢ Δᵢ ≤ c(m, n) . In addition, we prove an unconditional and completely explicit version of this result when $(m, n) = (1, 3)$ and thereby determine all the triples (x₁, x₂, x₃) of singular moduli such that a₁ x₁ + a₂ x₂ + a₃ x₃ ∈ Q for some a₁, a₂, a₃ ∈ Q \0\ .